Note that by (REF ) and Lemma REF , we have \(i_{{\omega }} (\xi _{{\alpha }_{k},e}) = 0\) .
Suppose \(b_2-b_3 =0\) , by Lemma 1.9.2 in p. 43 of [1]},
\(\nu _{{\omega }} (N_2({\omega },b)) =2\)  and then \(N_2({\omega },b)\)  has basic normal form \(R({\theta })\diamond R(2\pi -{\theta })\)  by the study in case 4
in p. 40 of [2]}.
Thus we have following contradiction.
\(0=i_{\omega }\left(\xi _{{\alpha }_{k}(e), e}\right)=i_{1}\left(\xi _{{\alpha }_{k}(e), e}\right)+S_{M}^{+}(1)-S_{R(\theta )}^{-}(\omega )-S_{R(\theta )}^{-}(\bar{\omega }) \le -1.\) 
